The myth that which is banned from integrated relatives

Nearly all American copy editors tend to have standing instructions
to change which to that if it introduces an integrated
relative clause. This is a tragic waste of many hundreds of people's
time. It has never been the case that which is forbidden, or
uncommon, in integrated relative clauses. It is quite uncommon
for ambiguity to arise over whether a relative clause is integrated
or supplementary, because flanking commas should always be used to mark
off the latter; so insisting that only supplementary relatives are
allowed to begin with which does hardly any work in reducing ambiguity.

Here is a list of a few Language Log posts discussing the issue
(these were written for fun more than scholarship; the tone, as usual
on Language Log, ranges from light-hearted detachment through stern
criticism to mock rage):

Those interested in the statistics on using which vs
that in integrated ("restrictive") relative clauses can find
some figures in print, in the Longman Grammar of Spoken and Written
English by Douglas Biber and colleagues. Page 616 provides some
figures for relative clauses in American and British newpapers (in
approximate numbers of occurrences per million words):

AmE news

BrE news

integrated relatives with which

800

2600

integrated relatives with that

3400

2200

supplementary relatives with which

1400

1400

supplementary relatives with that

0

0

The striking figure is the last one: virtually no supplementary
relatives are found with initial that any more. Actually such
examples do occur occasionally (though the figures above were based
on texts that happened not to contain any). Genuine supplementary
relatives can be identified unambiguously when the head noun is one
that doesn't take integrated relatives at all, like a proper name
or a similarly uniquely referring definite NP. So the sentence
His heart, that had lifted at the sight of Joanna, had become
suddenly heavy is an example of a supplementary relative beginning
with that. But such examples are extremely rare. See
this
Language Log post for the story of one confirmed sighting of this rare
species in the wild.

On the choice between that and which in integrated
relatives, there is a clear frequency difference between the dialect
groups — Americans tend to use which less; but reading
a broad range of English texts makes it obvious that both that
and which are grammatical in integrated relatives in both dialect
groups. Biber et al. speculatively attribute the difference to cultural
style: a greater "willingness to use a form with colloquial associations"
among Americans. It is not clear that there is really any support for
this. It seems more likely to be a response to what so many prescriptive
books still (wrongly) tell them.

What is clear is that the prescriptivists are simply wrong about
what should be prescribed: which relatives are common, both
from Americans (think of President Roosevelt's a day which will
live in infamy, uttered the day after the attack on Pearl Harbor)
and British writers ("It is the greatest blow which has befallen me
in my career", from The Hound of the Baskervilles by Sir
Arthur Conan Doyle, or "Never use a metaphor, simile or other figure
of speech which you are used to seeing in print" from George Orwell).
Thousands of examples from competent and even great writers could be
cited from either side of the Atlantic.